{"paper":{"title":"Phase transition in loop percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Art\\\"em Sapozhnikov, Yinshan Chang","submitted_at":"2014-03-22T17:01:15Z","abstract_excerpt":"We are interested in the clusters formed by a Poisson ensemble of Markovian loops on infinite graphs. This model was introduced and studied in [LeJ12] and [LL12]. It is a model with long range correlations with two parameters $\\alpha$ and $\\kappa$. The non-negative parameter $\\alpha$ measures the amount of loops, and $\\kappa$ plays the role of killing on vertices penalizing ($\\kappa>0$) or favoring ($\\kappa<0$) appearance of large loops. It was shown in [LL12] that for any fixed $\\kappa$ and large enough $\\alpha$, there exists an infinite cluster in the loop percolation on $\\mathbb{Z}^d$. In t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.5687","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}